Talk:Extended Cascading-E Notation
Has anyone studied this notation, and checked whether it actually has growth rate comparable to \(f_{\varphi(\omega,0,0)}(n)\)? I'm a little leery of his equations involving Knuth arrows on ordinals, and statements like "\(\omega\uparrow\uparrow\uparrow\omega = \Gamma_0\)" don't seem likely. But it's still possible that the notation has that growth rate; it's just that the Knuth arrows on ordinals is not a persuasive argument. Deedlit11 (talk) 13:42, March 21, 2014 (UTC) :How about this? Wythagoras (talk) 16:24, March 21, 2014 (UTC) :Our community has had some historical spats over how ordinals interact with arrow notation, and by extension, BEAF. This page accurately reflects the agreement we've reached (unless I screwed up the formula somehow, which is likely). Saibian's method is quite different, and I'm also a little skeptical about it. FB100Z • talk • 21:19, March 21, 2014 (UTC) From my own analysis, I can confirm that #^## -> \(\zeta_0\). I'm not sure beyond that though.-SJ224 14:07, November 5, 2014 (UTC) : Sorry, I meant #^^##. -SJ224 17:24, November 5, 2014 (UTC) I think \omega \uparrow\uparrow (\omega+1) cannot be \varepsilon_\omega because we can't "reach" other end of true infinity with "+1". But Saibian's #-towers, as well as Bowerian X's aren't supposed to be infinite, as far as I know. Ikosarakt1 (talk ^ ) 04:44, November 6, 2014 (UTC) :first sentence makes no sense it's vel 08:23, November 6, 2014 (UTC) I'll admit the approach the notation uses is somewhat unorthodox, but nonetheless, it's still the approach the notation uses. It still took me a while to understand it though. -SJ224 11:18, November 6, 2014 (UTC) What Ikosarakt1 said in his first sentence is perfectly clear in context. It's in response to the theory underwhich I constructed xE^ mentioned in the preliminary writings before the actual definition is given. That theory is in essence that the fundamental sequence of e1 = {e0+1,w^(e0+1),w^w^(e0+1),w^w^w^(e0+1),...}, implies that there is a "+1" that is climbing an infinite tower of "w"s, just as Bowers' suggests in his hypernomials page. The conclusion that xE^ is base on is that when we take the supremum ''of this we get something which can only be described as w^2#w. In otherwords, w^w^w^w^ ... ^2 w/w w's. Why 2? Because e0 can be thought of as w^w^w^w^w^...^1 w/w w's. So if the "+1" climbed to the top, the theory goes, you would get w^w^w^w^...^(1+1) = w^w^w^w^...^2 w/w w's. Perhaps Iko's point is much clearer now. Whether this theory is sound or not however is kind of irrelevant, because xE^ still defines fundamental sequences for everything. #^^## is still an ordinal notation even if it doesn't correspond to the idea of w^^(w^2) and even if that idea "makes no sense". Furthermore, because this theory ''is ''based on X structures, one objection that can be raised to Iko's point is that #^^# should NOT be thought of as infinitely tall. Rather it's simply as tall as whatever number is choosen for the keystone hyperion, which is strictly finite. Thus we can eventually climb the tower for any given value of #. This is the thinking that went behind xE^ and it was heavily inspired by Bowers' own ideas of the so called climbing method. Sbiis Saibian (talk) 16:10, November 6, 2014 (UTC) Note Under these rules (because rule 2 applies before rule 3, doesn't it?) , E100#^#1 = E100#100, which is grangol. Similiar, E100#^##1 = E100#^#100 = godgahlah = E100#^^#1. Wythagoras (talk) 17:46, November 6, 2014 (UTC) Almost all correct, but E100#^^#1 = E100#100 = ''grangol, and E100#^^#2 = E100#^#100 = godgahlah. Although it seems weird, having a 1 at the end doesn't necessarily mean we chop off the last delimiter and argument. This only happens if the delimiter is the proto-hyperion or if it's a successor-type. If it's a limit-type, it is more convenient to simply define a fundamental sequence which includes a "1-state". These can collapse surprisingly fast even for very large limit-delimiters. For example E100{#,#,1,2}1 = E100#100 = grangol, yet E100{#,#,1,2}2 = E100#{#}#100 = E100#{100}#100 = godsgodgulus ''More complex delimiters will not collapse so rapidly. For example E100#^(#^#*#)1 = E100#^#^#100 = ''godgathor. Sbiis Saibian (talk) 23:41, November 6, 2014 (UTC) &decomp What is the correct name for the set of decomposable-delimiters: a) “&decomp”; or b) “&decomp;”? -- 17:39, May 5, 2015 (UTC) / I attempted to create a redirect there, but this was blocked by a spam filter. Can somebody else create the redirect from / to Extended Cascading-E notation#Further extensions? -- 21:07, February 26, 2018 (UTC)